Differentiation of trigonometric functions
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable.
For example, the derivative of the sine function is written sin′(a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle.
All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as tan(x) = sin(x)/cos(x).
Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation.
The diagram at right shows a circle with centre O and radius r = 1.
Let two radii OA and OB make an arc of θ radians.
In the diagram, let R1 be the triangle OAB, R2 the circular sector OAB, and R3 the triangle OAC.
The area of triangle OAB is: The area of the circular sector OAB is: The area of the triangle OAC is given by: Since each region is contained in the next, one has: Moreover, since sin θ > 0 in the first quadrant, we may divide through by 1/2 sin θ, giving: In the last step we took the reciprocals of the three positive terms, reversing the inequities.
For the case where θ is a small negative number –1/2 π < θ < 0, we use the fact that sine is an odd function: The last section enables us to calculate this new limit relatively easily.
In this calculation, the sign of θ is unimportant.
Using cos2θ – 1 = –sin2θ, the fact that the limit of a product is the product of limits, and the limit result from the previous section, we find that: Using the limit for the sine function, the fact that the tangent function is odd, and the fact that the limit of a product is the product of limits, we find: We calculate the derivative of the sine function from the limit definition: Using the angle addition formula sin(α+β) = sin α cos β + sin β cos α, we have: Using the limits for the sine and cosine functions: We again calculate the derivative of the cosine function from the limit definition: Using the angle addition formula cos(α+β) = cos α cos β – sin α sin β, we have: Using the limits for the sine and cosine functions: To compute the derivative of the cosine function from the chain rule, first observe the following three facts: The first and the second are trigonometric identities, and the third is proven above.
Using these three facts, we can write the following, We can differentiate this using the chain rule.
, we have: Therefore, we have proven that To calculate the derivative of the tangent function tan θ, we use first principles.
By definition: Using the well-known angle formula tan(α+β) = (tan α + tan β) / (1 - tan α tan β), we have: Using the fact that the limit of a product is the product of the limits: Using the limit for the tangent function, and the fact that tan δ tends to 0 as δ tends to 0: We see immediately that: One can also compute the derivative of the tangent function using the quotient rule.
The numerator can be simplified to 1 by the Pythagorean identity, giving us, Therefore, The following derivatives are found by setting a variable y equal to the inverse trigonometric function that we wish to take the derivative of.
Using implicit differentiation and then solving for dy/dx, the derivative of the inverse function is found in terms of y.
To convert dy/dx back into being in terms of x, we can draw a reference triangle on the unit circle, letting θ be y.
Using the Pythagorean theorem and the definition of the regular trigonometric functions, we can finally express dy/dx in terms of x.
We let Where Then Taking the derivative with respect to
in from above, We let Where Then Taking the derivative with respect to
We let Where Then Taking the derivative with respect to
Then Taking the derivative with respect to
is derived as shown above, then using the identity
Let Then (The absolute value in the expression is necessary as the product of secant and tangent in the interval of y is always nonnegative, while the radical
is always nonnegative by definition of the principal square root, so the remaining factor must also be nonnegative, which is achieved by using the absolute value of x.)
Let Where Then, applying the chain rule to
: Let Then (The absolute value in the expression is necessary as the product of cosecant and cotangent in the interval of y is always nonnegative, while the radical
is always nonnegative by definition of the principal square root, so the remaining factor must also be nonnegative, which is achieved by using the absolute value of x.)
Let Where Then, applying the chain rule to
